Subgroups — C₂²×C₃×C₂²⋊C₄ #5842

Order 192.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₂²⋊C₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	—	2	8	no	—
6	C₂⁴×C₃	C₄	1	8	yes	complemented
7	C₂⁴×C₃	C₂²	1	6	yes	—
8	C₂⁴×C₃	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄	—	2	8	no	—
10	C₂²×C₃×C₄	C₂²	1	12	yes	complemented
11	C₃×C₂²⋊C₄	C₂²	1	16	yes	direct product factor
12	C₂⁵	C₂×C₃	1	1	yes	characteristic
13	C₂³×C₄	C₂×C₃	1	2	yes	complemented
14	C₂×C₂²⋊C₄	C₂×C₃	1	12	yes	direct product factor
15	C₂³×C₃	—	2	32	no	—
16	C₂³×C₃	—	2	24	no	—
17	C₂³×C₃	C₂×C₄	1	4	yes	—
18	C₂³×C₃	C₂×C₄	1	24	yes	complemented
19	C₂³×C₃	D₄	1	8	yes	—
20	C₂³×C₃	C₂³	1	4	yes	—
21	C₂³×C₃	C₂³	1	3	yes	—
22	C₂×C₃×C₄	—	2	24	no	—
23	C₂×C₃×C₄	C₂³	1	8	yes	complemented
24	C₂⁴	—	2	8	no	—
25	C₂⁴	C₃×C₄	1	8	yes	complemented
26	C₂⁴	C₂²×C₃	1	6	yes	—
27	C₂⁴	C₂²×C₃	1	1	yes	characteristic
28	C₂²×C₄	—	2	8	no	—
29	C₂²×C₄	C₂²×C₃	1	12	yes	complemented
30	C₂²⋊C₄	C₂²×C₃	1	16	yes	direct product factor
31	C₂²×C₃	—	2	48	no	—
32	C₂²×C₃	—	2	8	no	—
33	C₂²×C₃	C₂²⋊C₄	1	16	yes	direct product factor
34	C₂²×C₃	C₂²×C₄	1	6	yes	—
35	C₂²×C₃	C₂²×C₄	1	8	yes	complemented
36	C₂²×C₃	C₂×D₄	1	12	yes	—
37	C₂²×C₃	C₂⁴	1	1	yes	characteristic
38	C₃×C₄	—	2	8	no	—
39	C₂³	—	2	32	no	—
40	C₂³	—	2	24	no	—
41	C₂³	C₂×C₃×C₄	1	4	yes	—
42	C₂³	C₂×C₃×C₄	1	24	yes	complemented
43	C₂³	C₃×D₄	1	8	yes	—
44	C₂³	C₂³×C₃	1	4	yes	—
45	C₂³	C₂³×C₃	1	3	yes	—
46	C₂×C₄	—	2	24	no	—
47	C₂×C₄	C₂³×C₃	1	8	yes	complemented
48	C₂×C₃	—	2	8	no	—
49	C₂×C₃	C₂×C₂²⋊C₄	1	12	yes	direct product factor
50	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
51	C₂×C₃	C₂²×D₄	1	2	yes	—
52	C₂²	—	2	48	no	—
53	C₂²	—	2	8	no	—
54	C₂²	C₃×C₂²⋊C₄	1	16	yes	direct product factor
55	C₂²	C₂²×C₃×C₄	1	6	yes	—
56	C₂²	C₂²×C₃×C₄	1	8	yes	complemented
57	C₂²	C₂×C₃×D₄	1	12	yes	—
58	C₂²	C₂⁴×C₃	1	1	yes	characteristic
59	C₄	—	2	8	no	—
60	C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
61	C₂	—	2	8	no	—
62	C₂	C₂×C₃×C₂²⋊C₄	1	12	yes	direct product factor
63	C₂	C₂³×C₃×C₄	1	1	yes	characteristic
64	C₂	C₂²×C₃×D₄	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂²
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂³×C₃×C₄	C₂	semidirect
3. C₂×C₃×C₂²⋊C₄	C₂	direct (N × H)
4. C₂²×C₂²⋊C₄	C₃	direct (N × H)
6. C₂⁴×C₃	C₄	semidirect
10. C₂²×C₃×C₄	C₂²	semidirect
11. C₃×C₂²⋊C₄	C₂²	direct (N × H)
13. C₂³×C₄	C₂×C₃	semidirect
14. C₂×C₂²⋊C₄	C₂×C₃	direct (N × H)
18. C₂³×C₃	C₂×C₄	semidirect
23. C₂×C₃×C₄	C₂³	semidirect
25. C₂⁴	C₃×C₄	semidirect
29. C₂²×C₄	C₂²×C₃	semidirect
30. C₂²⋊C₄	C₂²×C₃	direct (N × H)
33. C₂²×C₃	C₂²⋊C₄	direct (N × H)
35. C₂²×C₃	C₂²×C₄	semidirect
42. C₂³	C₂×C₃×C₄	semidirect
47. C₂×C₄	C₂³×C₃	semidirect
49. C₂×C₃	C₂×C₂²⋊C₄	direct (N × H)
54. C₂²	C₃×C₂²⋊C₄	direct (N × H)
56. C₂²	C₂²×C₃×C₄	semidirect
60. C₃	C₂²×C₂²⋊C₄	direct (N × H)
62. C₂	C₂×C₃×C₂²⋊C₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁵×C₃	2
C₂³×C₃×C₄	2
C₂×C₃×C₂²⋊C₄	2
C₂²×C₂²⋊C₄	3

Subgroups — C22×C3×C22⋊C4 #5842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₂²⋊C₄ #5842